V. L. Girko
The Strong Elliptical Galactic Law. Sand Clock density. Twenty years later. Part II
We prove the strong Elliptical Galactic law for random matrices Ξn of the general form,
i.e. their diagonal entries have nonzero expectations and the pairs of the entries ( , )
have nonzero covariances. In this case the Elliptical Galactic law means that the support of the
accompanying spectral density of eigenvalues of matrix Ξn looks like the picture of several galaxies
made by telescope:
The picture 1 shows the collision of elliptic supports of the limit spectral density of n.s.f. of random
matrix An + ΛnΞn, where An is a diagonal complex matrix with diagonal entries (0.7, 0), (−1, 0),
(0, 0.7i) for corresponding three equal parts of the main diagonal, and random matrix Ξn has equal
covariances ?(√? = 0.2 + i0.8) of independent pairs of entries ( , ) with zero mean and is
multiplied by diagonal matrix Λn with diagonal entries (1, 0), (0.5, 0.5i), (−1, 0) for corresponding
three equal parts of the main diagonal. We have chosen in picture 1 three different diagonal entries of
the matrix An at a short distance.
In picture 2, we consider the diagonal matrix An with diagonal entries (2, 0), (−2, 0), (0, 2i) at
a large distant for corresponding three equal parts of the main diagonal. In the letter case we have
several domains-supports like ellipses. For the exposition of the Elliptic Law we have chosen the random
matrix Ξn of dimension 30 and 300 its Monte-Carlo simulation.
If the distances between the centers of these galaxies are large enough we have several almost
elliptical galaxies.
These pictures show the elliptic support of the limit spectral density of n.s.f. of random matrix
An + Ξn, where An is a diagonal matrix with 5 different diagonal entries (1, 0); (−1, 0); (−0.5,−i); (0, 0.5i); (0, i) and random matrix Ξn has equal covariances ?(√? = 0.5 + i0.5)of the entries . We have chosen five different diagonal entries of the matrix An at a short distance in picture 1 and at a large distant (2, 0); (−2, 0); (−1,−2i); (0, i); (0, 2i) in picture 2. In the letter
case we have several domains-supports like ellipses. For the exposition of the Elliptic Law we have
chosen the random matrix Ξn of dimension 50 and 300 its Monte-Carlo simulation.
If the distances between the centers of these galaxies are large enough we have several almost
elliptical galaxies.
Maybe the reader remembers the Monte Carlo simulations of eigenvalues of matrices Ξn + An,
where Ξn belongs to the domain of attraction of Circular law and An is the diagonal matrix whose diagonal entries forms letter R on a complex plain[25]–[27]. For the case when the matrix Ξn belongs
to the domain of attraction of Elliptic law the simulation of eigenvalues of the matrix Ξn + An looks like the following picture:
These statements are based on the VICTORIA-transform of random matrix which is the abbreviation
of the following words: Very Important Computational Transformation Of Random Independent Arrays.
We follow the main strategy of the theory of limit theorems of the probability theory, i.e. we try to
solve the problem of description of all limits of normalized spectral functions
where λk(AnΞnBn + Cn) are eigenvalues of non Hermitian matrix AnΞnBn + Cn, An, Bn, and
Cn are nonrandom matrices, under general (as only possible) conditions on the entries of random
matrices Ξn, χ is the indicator function. We emphasize that the spectral theory of Hermitian random
matrices is rather profound theory [3,13,23,24].
There are essentially three methods of the proof of Elliptic Laws that have been proposed: the
REFORM method and Berry-Esseen inequality[11], the method of perpendiculars[15,16], the method
of the central limit theorem and limit theorems for eigenvalues of random matrices[23]. The main
advantage of REFORM approach is that it enables the results of the previous version of Elliptic law
to be extended to the case under consideration. The REFORM-method(or G-martingale approach)
enables us to suggest a new method for construction of stochastic canonical equations.
In this paper we prove the following Elliptical Galactic Law which generalizes the Strong Circular
Law and Weak Circular Law(see the sketch of the proof of this law in the paper V-transform, Dopovidi
Akademii nauk Ukrainskoi RSR, Seria A Fizyko-Matematychni ta technichni nauky, 1982, N3, pp.5-6.):
For every n, let the pairs of random entries of the complex matrix be independent and given on a common probability space,
i, j = 1, ..., n, and
where are square
complex nonrandom matrices, det An ≠ 0, det Bn ≠ 0, and the real and imaginary parts of random entries have the densities
satisfying the corrected Elliptic condition: for some β > 1
or
and there exist the densities of the random entries , or the densities of the random entries , satisfying the condition: for some β1 > 1
the Lyapunov condition is fulfilled: for some δ > 0,
Then, with probability one, for almost all x and y
where
λk are eigenvalues of the matrix AnΞnBn + Cn, the Global probability density pn,α(t, s) =(∂2/∂t ∂s)Fn,α(t, s) is equal to
where α > 0,
where is a block matrix,
are entries of the matrix
is the block diagonal matrices, whose diagonal block satisfy the system of canonical equations K97
, j = 1, ..., n, and is a support of the Global probability density, where
There exists a unique solution of canonical equation K97 in the class of positive definite block matrices of the order 2 × 2, analytic in y > 0, t, s.
Random Operators and Stochastic Equations, Walter de Gruyter
Print ISSN: 0926-6364
Volume: 14, 04/2006
Pages: 157 - 208
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